The asymptotic variance of departures in critically loaded queues

We consider the asymptotic variance of the departure counting process D(t) of the GI/G/1 queue; D(t) denotes the number of departures up to time t. We focus on the case that the system load rho equals 1, and prove that the asymptotic variance rate satisfies lim_t Var D(t)/t = lambda (1 - 2/pi) (c_a2 + c_s2) where lambda is the arrival rate and c_a2 and c_s2 are squared coefficients of variation of the inter-arrival and service times respectively. As a consequence, the departures variability has a remarkable singularity in case rho equals 1, in line with the BRAVO effect (Balancing Reduces Asymptotic Variance of Outputs) which was previously encountered in the finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multi-server queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue we present an explicit expression of the variance of D(t) for any t.